Change in $p$-value and the posterior of the null as the luminosity grows

Bayesian and frequentist approaches to resonance searches

Change in $p$-value and the posterior of the null as the luminosity grows

Bayesian and frequentist approaches to resonance searches

Abstract

We investigate Bayesian and frequentist approaches to resonance searches using a toy model based on an ATLAS search for the Higgs boson in the diphoton channel. We draw pseudo-data from the background only model and background plus signal model at multiple luminosities, from about 0 to 10${}^6$/fb. We chart the change in the Bayesian posterior of the background only model and the global p-value. We find that, as anticipated, the posterior converges to certainty about the model as luminosity increases. The p-value, on the other hand, randomly walks between 0 and 1 under the background only model. After briefly commenting on the frequentist properties of the posterior, we make a direct comparison of the significances obtained in Bayesian and frequentist frameworks. We find that the well-known look-elsewhere effect reduces local significances by about 1$\sigma$. We furthermore find a previously unknown effect: significances from our Bayesian framework are typically about 1 to 2$\sigma$ smaller than the global significances. This suggests that even global significances could significantly overstate the evidence against the background only model. This effect - the Bayes effect - could radically change our interpretation of the evidence for new physics in resonance searches. We checked that the effect was robust with respect to thirteen choices of prior.